Bài tập 57 trang 218 SGK Toán 10 NC

Bài tập 57 trang 218 SGK Toán 10 NC

a)

\(\begin{array}{l}
2\sin \left( {\frac{\pi }{4} + \alpha } \right)\sin \left( {\frac{\pi }{4} - \alpha } \right)\\
 = \cos 2\alpha  - \cos \frac{\pi }{2} = \cos 2\alpha 
\end{array}\)

b)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\sin \alpha \left( {1 + \cos 2\alpha } \right)\\
 = \sin \alpha \left( {1 + 2{{\cos }^2}\alpha  - 1} \right)
\end{array}\\
{ = 2\sin \alpha {{\cos }^2}\alpha  = \sin 2\alpha \cos \alpha }
\end{array}\)

c)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\frac{{1 + \sin 2\alpha  - \cos 2\alpha }}{{1 + \sin 2\alpha  + \cos 2\alpha }}\\
 = \frac{{\sin 2\alpha \left( {1 - \cos 2\alpha } \right)}}{{\sin 2\alpha \left( {1 + \cos 2\alpha } \right)}}
\end{array}\\
\begin{array}{l}
 = \frac{{\sin 2\alpha  + 2{{\sin }^2}\alpha }}{{\sin 2\alpha  + 2{{\cos }^2}\alpha }}\\
 = \frac{{2\sin \alpha \left( {\cos \alpha  + \sin \alpha } \right)}}{{2\cos \alpha \left( {\cos \alpha  + \sin \alpha } \right)}} = \tan \alpha 
\end{array}
\end{array}\)

d)

\(\begin{array}{l}
\tan \alpha  - \frac{1}{{\tan \alpha }} = 2.\frac{{{{\tan }^2}\alpha  - 1}}{{2\tan \alpha }}\\
 =  - \frac{2}{{\tan 2\alpha }}
\end{array}\)

-- Mod Toán 10